A331817 a(n) = (n!)^2 * Sum_{k=0..n} (2*k)! / (2^k * (k!)^3 * (n - k)!).

1, 2, 9, 66, 681, 9090, 148905, 2889810, 64805265, 1648535490, 46896669225, 1475099460450, 50831084252025, 1904311245686850, 77061447551313225, 3349828945512299250, 155672917524626126625, 7701743926471878533250, 404153655359180645543625

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..380

Formula

E.g.f.: 1 / sqrt(1 - 4*x + 3*x^2).

a(n) = Sum_{k=0..n} binomial(n,k)^2 * (2*k - 1)!! * (n - k)!.

a(n) = n! * 2F1(1/2, -n; 1; -2).

a(n) ~ 3^(n + 1/2) * n^n / exp(n). - Vaclav Kotesovec, Jan 28 2020

D-finite with recurrence a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n). - Robert Israel, Feb 17 2020

Maple

f:= gfun:-rectoproc({a(n + 2) = 2*(3 + 2*n)*a(n + 1) - 3*(n + 1)^2*a(n), a(0)=1, a(1)=2}, a(n), remember):
map(f, [$0..30]); # Robert Israel, Feb 17 2020

Mathematica

Table[n!^2 Sum[(2 k)!/(2^k k!^3 (n - k)!), {k, 0, n}], {n, 0, 18}]
nmax = 18; CoefficientList[Series[1/Sqrt[1 - 4 x + 3 x^2], {x, 0, nmax}], x] Range[0, nmax]!
Table[n! Hypergeometric2F1[1/2, -n, 1, -2], {n, 0, 18}]

Prog

(PARI) seq(n) = {Vec(serlaplace(1/(sqrt(1 - 4*x + 3*x^2 + O(x*x^n)))))} \\ Andrew Howroyd, Jan 27 2020
(Magma) [(Factorial(n))^2*&+[Factorial(2*k)/(2^k*(Factorial(k))^3*Factorial(n-k)):k in [0..n]]:n in [0..18]]; // Marius A. Burtea, Jan 27 2020

Crossrefs

Cf. A000165, A001147, A032031, A034430, A052577.

Sequence in context: A214930 A089471 A196193 * A118804 A365995 A020555

Adjacent sequences: A331814 A331815 A331816 * A331818 A331819 A331820

Keyword

nonn

Author

Ilya Gutkovskiy, Jan 27 2020

Status

approved